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Physics
E. M. GANAPOL’SKII, A. N. CHERNETS
EXCITATION OF HYPERSOUND BY SLOW ELECTROMAGNETIC WAVES
(Presented by Academician N. N. Andreev, 12 IX 1962)
- Recently, papers have appeared in the literature devoted to the excitation of hypersonic waves. This is connected with the fact that the study of the absorption of hypersound in a solid at the frequencies of thermal phonons opens up the possibility of obtaining important information on the spin-lattice interaction in paramagnetic crystals and on the electron-phonon interaction in semiconductors and superconductors (¹).
The method currently used for exciting hypersound is based on the piezoelectric effect in a thin quartz rod (², ³) or bar (⁴) placed in the electric field of a cavity resonator. In this way hypersonic waves with a frequency of \(2.4 \cdot 10^{10}\) Hz were obtained (⁵). The use of this method for obtaining substantially higher hypersound frequencies encounters serious difficulties, connected mainly with the reduction of the dimensions and the quality factor of the resonator.
Hypersonic waves are excited in quartz in a thin surface layer (²). This makes it possible, for obtaining hypersound, to use surface slow electromagnetic waves propagating in periodic structures. Such a method of exciting hypersound proves to be very effective and makes it possible to attain higher frequencies.
- Let us consider a quartz single crystal bounded by a plane, near which a surface slow electromagnetic wave propagates. We choose the direction of the coordinate axes \(x, y, z\) along the crystallographic directions in quartz \(X, Y, Z\), respectively. The plane bounding the quartz is \(x = 0\). The projections of the electric-field strength vector in a surface electromagnetic wave propagating in the \(y\) direction are:
\[ E_x = E_{0x} e^{-px+ihy-i\omega t}; \qquad E_y = E_{0y} e^{-px+ihy-i\omega t}; \qquad E_{0y} = \frac{ip}{h} E_{0x}; \qquad E_z = 0, \tag{1} \]
where \(\omega, h\) are the frequency and wave number; \(p^2 = h^2 - k_0^2\); \(k_0 = c\omega\); \(c\) is the speed of light.
According to (⁶), in the case of quartz the equations of motion will have the following form:
\[ \begin{aligned} \rho \ddot{u}_x &- \lambda_{1111}\frac{\partial^2 u_x}{\partial x^2} - \lambda_{1212}\frac{\partial^2 u_x}{\partial y^2} - \frac{\partial}{\partial x \partial y} \left[(\lambda_{1122}+\lambda_{1212})u_y+(\lambda_{1132}+\lambda_{1231})u_z\right] \\ &= \beta_{111}\frac{\partial E_X}{\partial x} + \beta_{212}\frac{\partial E_Y}{\partial y}; \tag{2} \\[1em] \rho \ddot{u}_y &- \frac{\partial^2}{\partial x^2}(\lambda_{2113}u_z+\lambda_{2112}u_y) - \frac{\partial^2}{\partial y^2}(\lambda_{2222}u_y+\lambda_{2223}u_z) \\ &- \frac{\partial}{\partial x \partial y} \left[(\lambda_{2112}+\lambda_{1122})u_x\right] = \beta_{221}\frac{\partial E_Y}{\partial x} + \beta_{122}\frac{\partial E_X}{\partial y}; \\[1em] \rho \ddot{u}_z &- \frac{\partial^2}{\partial x^2}(\lambda_{3113}u_z+\lambda_{3112}u_y) - \frac{\partial^2}{\partial y^2}(\lambda_{3222}u_y+\lambda_{3223}u_z) \\ &- \frac{\partial}{\partial x \partial y} \left[(\lambda_{3112}+\lambda_{3211})u_x\right] = \beta_{231}\frac{\partial E_Y}{\partial x} + \beta_{132}\frac{\partial E_X}{\partial y}. \end{aligned} \]
Boundary conditions at \(x=0\)
\[ \frac{\partial u_x}{\partial x}=\gamma_{111}E_X;\qquad \frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}=2\gamma_{221}E_Y;\qquad \frac{\partial u_z}{\partial x}=2\gamma_{231}E_Y. \tag{3} \]
Owing to the symmetry of the problem,
\[ \frac{\partial u_x}{\partial z}=\frac{\partial u_y}{\partial z}=\frac{\partial u_z}{\partial z}=0, \]
where \(u_x, u_y, u_z\) are the projections of the deformation vector; \(\lambda_{iklm}\) is the tensor of elastic moduli; \(\beta_{l,ik}\), \(\gamma_{l,ik}\) are the piezoelectric tensors; \(\beta_{l,ik}=\gamma_{l,mn}\lambda_{mn,ik}\); \(\rho\) is the density of quartz.
Separating out the small parameter \(\mu=h/k_1=v_s^{(1)}/v_e\)—the ratio of the speed of sound in quartz to the speed of the surface electromagnetic wave—and neglecting terms of order higher than \(\mu\), we obtain the solution of the system of equations (2) satisfying the boundary conditions (3):
\[ \begin{aligned} u_x&=u_{0x}P_1+O(\mu)P_2+O(\mu)P_3+p_xP_4;\\ u_y&=O(\mu)P_1+\alpha u_{0y}P_2+\beta u_{0z}P_3+p_yP_4;\\ u_z&=O(\mu)P_1+\beta u_{0y}P_2-\alpha u_{0z}P_3+p_zP_4.\\ P_i&=e^{ik_i x+ihy-i\omega t},\quad i=1,2,3;\qquad P_4=e^{-px+ihy-i\omega t},\\ \alpha^2+\beta^2&=1, \end{aligned} \tag{4} \]
where \(k_i=\omega/v_s^{(i)}\), \(v_s^{(i)}\) are the velocities of sound propagation in the direction of the \(x\)-axis; \(\alpha^2=(\lambda_{1313}-\rho v_s^{(2)2})[\lambda_{2113}^2+(\lambda_{3113}-\rho v_s^{(2)2})^2]^{-1}\).
The solution of the secular equation corresponding to (2) determines the dependence of \(v_s^{(i)}\) on the moduli \(\lambda_{iklm}\):
\[ (v_s^{(1)})^2=\frac{\lambda_{1111}}{\rho};\qquad \rho\,(v_s^{(2,3)})^2=\frac{1}{2}(\lambda_{2323}+\lambda_{1212})\pm \]
\[ \pm\left[\frac{1}{4}(\lambda_{2323}+\lambda_{1212})^2+\lambda_{1213}^2-\lambda_{1212}\lambda_{2323}\right]^{-2}. \tag{5} \]
The amplitudes of the hypersound waves, determined from the boundary conditions (3), are:
\[ u_{0x}=\gamma_{111}\frac{E_{0x}}{ik_1};\qquad u_{0y}=\frac{2E_{0y}}{ik_2}\left(\frac{\alpha\gamma_{221}+\beta\gamma_{231}}{\alpha^2+\beta^2}\right);\qquad u_{0z}=\frac{2E_{0y}}{ik_3}\left(\frac{\beta\gamma_{221}-\alpha\gamma_{231}}{\beta-\beta^2+\alpha^2}\right). \tag{6} \]
The quantities \(p_x, p_y, p_z\) are determined on the basis of the solution of the inhomogeneous equation (2)
\[ p_x=\mu\,\frac{(\beta_{111}E_{0x}-i\beta_{212}E_{0y})}{\lambda_{1111}k_1};\qquad p_y=\mu\,\frac{(\beta_{221}E_{0y}-i\beta_{122}E_{0x})}{\lambda_{1111}k_1}; \]
\[ p_z=\mu\,\frac{(\beta_{221}E_{0y}-i\beta_{122}E_{0x})}{\lambda_{1111}k_1}. \tag{7} \]
In the zeroth approximation (\(\mu=0\)) it follows from (4) that, when a surface slow electromagnetic wave propagates near the boundary of quartz, three types of sound waves are excited in the quartz: (a) a purely longitudinal wave propagating along the \(x\)-axis with velocity \(v_s^{(1)}\); (b) two purely transverse waves, polarized mutually perpendicular to each other and propagating in the same direction with velocities \(v_s^{(2)}\) and \(v_s^{(3)}\).
The sources of the longitudinal and transverse waves are, respectively, the normal and tangential components of the electric field at the quartz boundary. For \(\mu\ne 0\), the “purity” of the waves is violated; transverse and longitudinal deformation components appear, respectively, in the longitudinal and transverse waves. In this same approximation, an ultrafast surface sound wave is excited, propagating in the direction \(y\) with the velocity of the surface electromagnetic wave.
Powers of the longitudinal and transverse hypersonic waves
\[ W_n=\frac{1}{2}\lambda_{1111}\gamma_{111}^{2}E_{0x}^{2}v_s^{(1)}S, \]
\[ W_{t_1}=\frac{1}{2}E_{0y}^{2}\left(\frac{\alpha\gamma_{221}+\beta\gamma_{231}}{\alpha^2+\beta^2}\right)^2 (\lambda_{1212}\alpha^2+\lambda_{1213}\alpha\beta+\lambda_{1313}\beta^2)v_s^{(2)}S, \tag{8} \]
\[ W_{t_2}=\frac{1}{2}E_{0y}^{2}\left(\frac{\beta\gamma_{221}-\alpha\gamma_{231}}{\beta^2+\alpha^2}\right)^2 (\lambda_{1212}\beta^2-\lambda_{1213}\alpha\beta+\lambda_{1313}\alpha^2)v_s^{(3)}S; \]
\(S\) is the cross-sectional area of the hypersonic beam.
Fig. 1. 1 — quartz bar with dimensions \(10\times15\times20\) mm; 2 — waveguide supplying microwave energy; 3 — “comb” with parameters \(\beta_e=0.05\), \(L=20\) mm
Power ratio
\[ W_n:W_{t_1}:W_{t_2}\simeq 1:\left(\frac{\rho}{h}\right)^2:\left(\frac{\rho}{h}\right)^2\,0.6. \tag{8a} \]
Using the Umov–Poynting theorem, for the conversion coefficient \(\eta_n\)—the ratio of the power of the longitudinal hypersonic wave to the power of the surface electromagnetic wave—we obtain
\[ \eta_n=\frac{w_n}{w_e} =\lambda_{1111}\gamma_{111}^{2}\frac{\lambda_s^{2}L\omega^2}{\pi\beta_e^2c^2}, \tag{9} \]
where \(\lambda_s\) is the wavelength of the sound wave, \(\beta_e\) is the slowing coefficient, \(\beta_e=v_e/c\); \(L\) is the length of the quartz along the \(y\)-axis.
Fig. 2. Oscillogram of pulses corresponding to longitudinal and transverse hypersonic waves at a frequency of \(4\cdot10^{10}\) Hz (the decreased spacing between the pulses on the oscillogram at the end of the sweep is due to its nonlinearity)
3. Using the method described, hypersound was excited in a quartz single crystal having the form of a rectangular parallelepiped whose edges are parallel to the crystallographic axes.
The faces of the parallelepiped perpendicular to the \(X\) axis were optically flat and parallel. To create the surface electromagnetic wave, a comb was used (Fig. 1). The block diagram of the apparatus is analogous to that described in \({}^{4}\).
In such a system, at the temperature of liquid helium (\(4.2^\circ\mathrm{K}\)), longitudinal and transverse hypersound waves with frequency \(4 \cdot 10^{10}\) Hz were excited. On the oscillogram (Fig. 2) three groups of pulses are visible: \((a_1 — a_4)\), \((b_1 — b_4)\), \((c_1 — c_2)\), corresponding to the longitudinal and two transverse waves propagating in the quartz in the direction of the \(X\) axis. The measured propagation velocities of the hypersound waves agree, within \(5\%\), with the values obtained at a frequency of \(10^{10}\) Hz \({}^{4}\). The ratio of the powers of the longitudinal and transverse waves agrees with \((8^a)\).
The results obtained show that the method of exciting hypersound by means of surface electromagnetic waves opens a real possibility of advancing to higher frequencies.
Institute of Radiophysics and Electronics
Academy of Sciences of the Ukrainian SSR
Received
12 IX 1962
CITED LITERATURE
\({}^{1}\) S. A. Altshuler, B. I. Kochelaev, A. M. Leushin, UFN, 75, 459 (1961).
\({}^{2}\) H. E. Bömmel, K. Dransfeld, Phys. Rev., 117, 1245 (1960).
\({}^{3}\) E. H. Jacobsen, Phys. Rev. Lett., 2, 249 (1959).
\({}^{4}\) E. M. Ganapolsky, A. N. Chernetz, ZhETF, 42, 12 (1962).
\({}^{5}\) E. H. Jacobsen, Proc. Intern. Conf. on Quantum Electronics, Sept. 1959.
\({}^{6}\) L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Moscow, 1959.